The Hot Hand

What Was Once a Fallacy

Alex Cardazzi

Old Dominion University

About Me

Completed my PhD at West Virginia University in 2021

Assistant Prof. of Economics at Old Dominion University

Probability is Tricky

“Monty Hall” problem

  1. Three doors, two goats (lose), one car (win).
  2. Contestant picks a door (suppose door #1).
  3. Monty Hall opens one of the doors with a goat behind it.

Source: Wikipedia

Probability is Tricky

“Monty Hall” problem

  1. Three doors, two goats (lose), one car (win).
  2. Contestant picks a door (suppose door #1).
  3. Monty Hall opens one of the doors with a goat behind it.
  4. Should the contestant keep their door or switch?

We can simulate the setting to find out the best strategy!

Monty Hall Simulation

Monty Hall Simulation

Monty Hall Explanation

Step 1: all three doors have a \(\frac{1}{3}\) probability of being the “winning” door.

Step 2: once the contestant select one door, this door has a \(\frac{1}{3}\) probability of being the winning door. The other two doors, in sum, have a \(\frac{2}{3}\) probability of being the winner.

Step 3: since Monty Hall opens one of the two doors, revealing nothing, the remaining door now has a higher chance of being the winner, conditional on not being the one opened.

Step 4: now, the chosen door has a \(\frac{1}{3}\) probability while the other door has a \(\frac{2}{3}\) probability. This is not to say that there is anything special about the door, but rather that the situation has altered the probabilities.

Another way to think about this: imagine there are 100 doors, you pick door #1, and Monty Hall opens every other door except for door #72.

Gilovich, Vallone, and Tversky (1985)

GVT (1985)

What is meant by the “Hot Hand”?

What is an example of the “Hot Hand”?

Who believes in the “Hot Hand”?

GVT (1985)

The “Hot Hand”

Also referred to as the “gambler’s fallacy,” this is the idea that certain outcomes are more or less likely following strings of successes or failures. Humans are great at identifying patterns, though this results in a high Type I error rate.

  1. Do fans believe in the hot hand?
    • 100 basketball fans from Cornell and Stanford were surveyed.
    • 91% believe in the hot hand
  2. Professional Field Goals (PHI 76ers)
  3. Professional Free Throws (BOS Celtics)
  4. Controlled Shooting Experiment\(^*\) (Cornell)
    • 14 men and 12 women, each taking 100 shots, no evidence
    • Allowed players (and observers) to “bet” on shot outcomes, no evidence
    • Did find that bets were correlated with previous shot outcome

The History of the Hot Hand

  • Lots of research in finance, economics, wagering, casino gambling, etc.
  • Camerer (1989): Does the (betting) market believe in the Hot Hand? TL;DR: Yes!
  • Koehler and Conley (2003) and Avugos, Bar-Eli, Ritov, and Sher (2013) replicate and confirm GVT.

… any further investment in a search for the hot hand is redundant …

Avugos et al. (2013)

  

Let’s be redundant, and run some simulations.

Hot Hand Simulation

Hot Hand Simulation

Using “Run Length Encoding” function (base::rle)

Hot Hand Simulation

Hot Hand Simulation

Hot Hand Simulation

Hot Hand Simulation

Hot Hand Simulation

Code
get_outcomez <- function(vec, n = 3){
  
  rle_vec <- rle(vec)
  df <- data.frame(length = rle_vec$lengths, value = rle_vec$values)
  df <- df[df$length >= n,]
  
  df$outcome <- 1 - df$value
  outcomez <- list()
  for(i in which(df$length > n)){
    
    df_mini <- df[rep(i, df$length[i] - n),]
    df_mini$outcome <- df_mini$value
    outcomez[[length(outcomez) + 1]] <- rbind(df_mini, df[i,])
  }
  
  outcomez[[length(outcomez) + 1]] <- df[which(df$length == n),]
  outcomez <- do.call(rbind, outcomez)
  return(outcomez)
}

Hot Hand Simulation

Hot Hand Simulation

Hot Hand Simulation

Explanation?

Suppose a researcher flips 100 coins and writes down the result of the flip after a streak of three heads. Each of these results are then put into a bag to look at later.

  • \(P(F_{42} = H) = \frac{1}{2}\)
  • \(P(F_{42} = H | F_{39} = F_{40} = F_{41} = H) = \frac{1}{2}\)
    • This is known because we are flipping coins.
  • If \(F_{42} = H\), the researcher now has \(F_{42} = H\) and (at least) \(F_{43}\) in her bag.
  • If \(F_{42} = T\), the researcher only has \(F_{42} = T\) in her bag.

The bias is due to sample selection, just like in Monty Hall.

Flipping Coins \(\rightarrow\) Basketball

This bias is in no way intuitive, but neither is Monty Hall problem. Now that we have identified a bias in the baseline calculation, we can re-do the original calculations with this in mind. So, what do Miller & Sanjurjo (2018) find after correcting?

GVT’s modest (and insignificant) 3% increase is in fact a 13% increase.

To put the magnitude of +13 percentage points into perspective, the difference between the median three point shooter and the top three point shooter in the 2015–2016 NBA season was 12 percentage points.

Moreover, these estimates are conservative – players often take more difficult shots as they “get hot.” Lastly, the bias found decreases as sequence length increases, but increases as streak length increases.

Flipping Coins \(\rightarrow\) Basketball

Miller & Sanjurjo propose alternatives to GVT:

Hit Streak Frequency – \(H_F = \frac{\#S_H}{\#S}\)

  • Num. shots that follow a streak (\(S_H\)) per number of shots (\(S\)). This measures how often a player is hot.

Hit Streak Momentum – \(H_M = \frac{\sum_{s \in S_H} x_s}{\#S_H}, \ \text{if} \ S_H > 0\)

  • Given shots that follow streaks, what is the player’s shooting %?

The exact distribution of each of these test statistics, under the null hypothesis that a player has a fixed probability of success, can be approximated to arbitrary precision with a Monte-Carlo permutation test.

Flipping Coins \(\rightarrow\) Basketball

  1. These statistics allow for testing “hot hand” and “cold hand” effects separately.

  2. We can compare a player’s hit streak momentum (\(H_M\)) to a null value (via simulation) rather than via their hit streak momentum following misses (which is biased!).

  3. Hit streak frequency (\(H_F\); and hit streak length) offer alternative definitions, and therefore tests, of the “Hot Hand.”

Create Null Distributions

Create Null Distributions

Comparing Sample Values to Null

Comparing Sample Values to Null

\(p\)-value

A \(p\)-value represents the probability of observing a sample value as extreme as, or more extreme than, the value observed, given the null hypothesis is true.

Post Miller & Sanjurjo (2018)

Miller & Sanjurjo (2021) examine perhaps the best “natural” shooting experiment: the NBA 3-Point Contest. The authors collect data from 34 years (1986–2020) and again find substantial evidence that points to players shooting better following streaks of makes.

Other research in sports has looked at in-game basketball shooting data, cricket, darts, golf, etc. Overturning such a consensus finding has led to an explosion of research on the Hot Hand.

… any further investment in a search for the hot hand is redundant …

Avugos et al. (2013)

Conclusion

To summarize:

  • Probability is hard, and humans are bad at it.
  • Swap doors next time you find yourself on a game show.
  • Believe in the “Hot Hand” – it’s real!
  • Simulate!

fin